Video: CBSE Class X • Pack 2 • 2017 • Question 9

CBSE Class X • Pack 2 • 2017 • Question 9

04:17

Video Transcript

A line intersects the 𝑦-axis and the 𝑥-axis at the points 𝑃 and 𝑄 respectively. If two, negative five is the midpoint of 𝑃𝑄, find the coordinates of 𝑃 and 𝑄.

Here’s our 𝑦-axis and our 𝑥-axis. We have a point two, negative five. We know that this point falls on our line and is in fact midway between 𝑃 and 𝑄. If we sketch an outline of what we think our line will look like, it could be something like this. We know that 𝑃 is the place where this line crosses the 𝑦-axis. And 𝑄 is the place where this line crosses the 𝑥-axis. And this point — two, negative five — should be halfway between them.

The midpoint of a line segment is found by taking the 𝑥-coordinates of the endpoints, dividing them by two, and then taking the 𝑦-coordinates of the end points and dividing by two. Our midpoint is two, negative five. Which means, 𝑥 one plus 𝑥 two divided by two equals two. And 𝑦 one plus 𝑦 two divided by two equals negative five. We can break this up into two separate equations. 𝑥 one plus 𝑥 two divided by two equals two. 𝑦 one plus 𝑦 two divided by two equals negative five.

From there, we need to label our 𝑃 and 𝑄. We can call 𝑃 𝑥 one, 𝑦 one and 𝑄 𝑥 two, 𝑦 two. Is there anything we can already say about the coordinates of 𝑃? We know that point 𝑃 intersects the 𝑦-axis. Which means, the 𝑥-coordinate of our 𝑃 value equals zero. We could label point 𝑃 as zero, 𝑦 one. Similarly, point 𝑄 crosses the 𝑥-axis. Which means, it has a 𝑦-coordinate of zero. So we’ll label point 𝑄 𝑥 two, zero.

Now, we plug in what we know into the formula. Starting with our 𝑥-values, 𝑥 one is on the 𝑃-coordinate. And we know that that equals zero. 𝑥 two is still unknown. And we bring everything else down. Zero plus 𝑥 two divided by two equals two. Zero plus anything is itself. And that means 𝑥 two divided by two equals two. We multiply by two over one on the left side and the right side. On the left, everything cancels. And we have 𝑥 two equals two times two. 𝑥 two equals four. The 𝑥 two coordinate is the value of 𝑥 for point 𝑄. 𝑄 must be located at four, zero.

We’ll repeat this process to find out what the 𝑦 one value is. 𝑦 one is unknown. 𝑦 two is the 𝑦-coordinate for point 𝑄. And we know that’s zero. We can say that 𝑦 one plus zero divided by two has to equal negative five. 𝑦 one plus zero equals 𝑦 one. 𝑦 one divided by two equals negative five.

Again, we need to multiply by two over one on the left and the right. On the left, we’re left with 𝑦 one equals negative five times two. 𝑦 one equals negative 10. This tells us that our point 𝑃 is located at negative 10 on the 𝑦-axis. Our question is specifically asking us to list the coordinates of 𝑃 and 𝑄. 𝑃 is located at zero, negative 10. And 𝑄 is located at point four, zero.