Question

# Tangents are drawn from the point $$\left(3, 2\right)$$ to the ellipse $${x}^{2}+4{y}^{2}=9$$. Find the equation to their chord of contact and the equation of the straight line joining $$\left(3, 2\right)$$ to the middle point of this chord of contact.

A
3y=2x
B
y2x=0
C
2x+y=3
D
None of the above

Solution

## The correct option is C $$3y=2x$$Given equation of ellipse is $${ x }^{ 2 }+4{ y }^{ 2 }=9$$ Equation of chord of contact is $$T=0$$ $$x{ x }_{ 1 }+4y{ y }_{ 1 }=9$$ $$x(3)+4y(2)=9$$ $$3x+8y=9$$    ......(i) Let the middle point of chord of contact be $$(h,k)$$ Equation of chord of contact when mid point is given is $$T=S'$$ $$hx+4ky={ h }^{ 2 }+{ k }^{ 2 }$$    .....(ii) Now (i) and (ii) are equation of same line  $$\dfrac { h }{ 3 } =\dfrac { 4k }{ 8 } =\dfrac { { h }^{ 2 }+{ k }^{ 2 } }{ 9 }$$$$\Rightarrow 3h={ h }^{ 2 }+{ k }^{ 2 }$$    ......(iii)$$\Rightarrow 2h=3k$$    ......(iv) Substituting (iv) in (iii), we get $$27h=9{ h }^{ 2 }+4{ h }^{ 2 }\\ \Rightarrow 13{ h }^{ 2 }-27h=0\\ \Rightarrow h=0,\dfrac { 27 }{ 13 }$$If $$h=0$$, then $$k=0$$  So, the middle point of chord of contact is $$(0,0)$$ Equation of line joining $$(0,0)$$ and $$(3,2)$$ is $$y-0=\dfrac { 2-0 }{ 3-0 } (x-0)\\ \Rightarrow 3y=2x$$ Hence, proved.Maths

Suggest Corrections
Â
0

Similar questions
View More

People also searched for
View More