Choose the Correct options
0 of 25 Questions completed
Questions:
You have already completed the quiz before. Hence you can not start it again.
Quiz is loading…
You must sign in or sign up to start the quiz.
You must first complete the following:
0 of 25 Questions answered correctly
Your time:
Time has elapsed
You have reached 0 of 0 point(s), (0)
Earned Point(s): 0 of 0, (0)
0 Essay(s) Pending (Possible Point(s): 0)
Average score 

Your score 

If a hyperbola passes through the point \( P (\sqrt{2}, \sqrt{3}) \) and has foci at \( (\pm 2 , 0) \), then the tangent to this hyperbola at \( P \) also passes through the point.
Let, (Equation of hyperbola)
Since lie on hyperbolaa
Now,
Equation of tangent at is given by which passes through
Let eccentricity of the hyperbola \( \dfrac{x^2}{a^2}\dfrac{y^2}{b^2}=1 \) be reciprocal to that of the ellipse \( x^2+4y^2=4 \), if the hyperbola passes through a focus of the ellipse, then_____
Hence, (Equation of the ellipse)
and focus
For hyperbola where,
and hyperbola passes through
From above two equations , we get
Equation of hyperbola is
Let \( P (6, 3) \) be a point on the hyperbola \(\dfrac{x^2}{a^2}\dfrac{y^2}{b^2}=1 \). If the normal at the point \( P \) intersects the \( x \)axis at \( (9, 0) \), then \( e \) is______
Equation of normal hyperbola at is
At
It passes through
If the line \( 2x+ \sqrt{6}y=2 \) touches the hyperbola \( x^22y^2=4 \), then the point of contact is ______
The equation of tangent at is , which is same as
and
Thus, the point of contact is
Let \( P (a \sec \theta, b \tan \theta) \) and \( Q (a \sec \phi, b \tan \phi) \), where \( \theta + \phi =\dfrac{\pi}{2} \), be two points on the hyperbola \( \dfrac{x^2}{a^2}\dfrac{y^2}{b^2}=1 \). If \((h, k) \) is the point of intersection of the normals at \( P \) and \( Q \), then \( k \) is______
Slope of normal to at
Slope of normal at is
Equation of normal is
(i)
Similarly, (ii)
we get
If \( 2xy+1 =0 \) is tangent to the hyperbola \(\dfrac{x^2}{a^2}\dfrac{y^2}{16}=1 \), then which of the following is correct answer to find \( a \)?
Tangent :
Hyperbola :
Parametric form :
tangent :
We get
Tangents are drawn to the hyperbola \( \dfrac{x^2}{9}\dfrac{y^2}{4}=1 \), parallel to the line \( 2xy=1 \). The point of contact of the tangent to the hyperbola is_____
Equation of tangent, parallel to is
Equation of tangent at at from above two equations, we get
The circle \( x^2+y^28x=0 \) and hyperbola \(\dfrac{x^2}{9}\dfrac{y^2}{4}=1 \) intersect at the points \( A \) and \( B \), then equation of the circle with \( AB \) as its diameter is ________
Given, Circle : , Hyperbola :
For their intersection,
(is not acceptable)
For
Required equation is
If the circle \( x^2+y^2=8x \) and hyperbola \(\dfrac{x^2}{9}\dfrac{y^2}{4}=1 \), intersect at the point \( (x ,y) \) then equation of a common tangent with positive slope to the circle as well as to the hyperbola is______
Equation of tangent to hyperbola having slope is (i)
Equation of tangent to circle is (ii)
Equation (i) and (ii) are identical for
Equation of common tangent is
The line \(2x+y=1 \) is tangent to the hyperbola \( \dfrac{x^2}{a^2}\dfrac{y^2}{b^2}=1 \). If the line passes through the point of intersection of the nearest directrix and the \(x \)axis, then the eccentricity of the hyperbola is_____
Given,
Substituting in , we get
Also, is tangent to hyperbola.
The conic is given hyperbola so
If \(x=9 \) is the chord of contact of the hyperbola \(x^2y^2=9 \), then the equation of the corresponding pair of tangents is_____
Let be a point whose chord of contact w.r.t hyperbola is
But it is equation of line ,
This is possible when
Again equation of pair of tangents is
If the circle \( x^2+y^2=a^2 \), intersects the hyperbola \(xy=c^2 \) in four points \( P(x_1 ,y_1) \), \( Q (x_2,y_2), \quad R (x_3,y_3), \quad S (x_4, y_4) \) then_____
Given,
We obtain,
Now, are roots of above equation then (sum of roots)
Tangents are drawn from any point on the hyperbola \(\dfrac{x^2}{9}\dfrac{y^2}{4}=1 \) to the circle \( x^2+y^2=9 \). Find the locus of midpoint of chord of contact.
Let any point on the hyperbola is
Chord of contact is
Equation of chord in midpoint form is
Locus:
If the ellipse with equation \( 9x^2+25y^2=225 \), then find \( e \)?
Given ,
Hence,
If the latus rectum of an ellipse is equal to half of the minor axis, then find its \( e \)?
(Equation of ellipse)
Length of major axis
Length of minor axis
Length of latus rectum
Given that,
If the eccentricity of the ellipse is \( \dfrac{5}{8} \) and the distance between its foci is \( 10 \), then find the latus rectum of the ellipse.
Given, (ellipse)
Distance between
x
length of latus rectum
Find the equation of ellipse whose eccentricity is \(\dfrac{2}{3} \), latus rectum is \( 5 \) and the centre is \((0, 0) \).
Given,
Equation of ellipse is
Find the distance between the directrices of ellipse \( \dfrac{x^2}{36}+\dfrac{y^2}{20}=1 \)
Given,
Now, directrices
Distance between the directrices
Find the coordinates of a point on the parabola \( y^2=8x \), whose focal distance is \( 4 \).
Given, , Hence
Focal distance
For
Coordinates of the points are and
Find the length of the line segment joining the vertex of the parabola \( y^2=4ax \) and a point on the parabola, where the line segment makes an angle \( \theta \) to the \(x \)axis.
Given,
In
If the points \((0, 4) \) and \( (0, 2) \) are respectively the vertex and focus of a parabola, then find the equation of the parabola?
Given, vertex and focus
If a line \( y=mx+1 \) is tangent to the parabola \( y^2 =4x \), then find the value of \( m \).
Given, (line) , (parabola)
From above equations
Find the equation of hyperbola, whose distance between the foci is \( 16 \) and eccentricity is \(\sqrt{2} \).
So equation of hyperbola is
Find the eccentricity of the hyperbola \( 9y^24x^2=36\)
Given,
Hence
Find the equation of the hyperbola with eccentricity \( \dfrac{3}{2} \) and foci at \( (\pm 2, 0) \)
Given, , foci
Equation of the hyperbola is