Question

Assertion :$\sqrt{(x-1{)}^2+y^2}+\sqrt{(x+1{)}^2+y^2}=4$ represents the equation of ellipse. Reason: The locus of point which moves in a plane such that sum of its distance from two fixed points is constant is an ellipse.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Given $\sqrt{(x-1{)}^2+y^2}+\sqrt{(x+1{)}^2+y^2}=4$
$\Rightarrow \sqrt{(x-1{)}^2+y^2}=4-\sqrt{(x+1{)}^2+y^2}$
Squaring both sides, we get
$(x-1{)}^2+y^2=16+(x+1{)}^2+y^2-8\sqrt{(x+1{)}^2+y^2}$
$\Rightarrow 4x+16=8\sqrt{(x+1{)}^2+y^2}$
Squaring both sides, we get
$64\left(\right(x+1{)}^2+y^2)=16(x+4{)}^2$
$\Rightarrow 3x^2+4y^2=12$
which is an ellipse.
Hence, assertion is correct.
For locus to be an ellipse, the sum of the distances from the two fixed points must be equal to the distance between the two points.
Hence, reason is correct.